Once I found out yesterday that I was denied for the CLI, I called the backdoor number and the account specialist told me that my card wasn't eligible. She also recommended that I apply for another Chase card.
I called customer service this morning and the representative told me that they DO NOT have cards that are not eligible for a CLI but would transfer me to a Credit Specialist. The credit specialist then told me that in fact my card was not eligible for "customer-initiated" CLI. I would be eligible for the auto-generated CLI. I asked her how long does that take and she said that it is auto.
I hope that answered your question FretlessMayhem